Viscous fingering of a thixotropic fluid in a porous medium or a narrow fracture

We consider the flow of a fluid with a modified Bautista–Manero rheology injected radially into a porous medium or unidirectionally into a widening fracture. Linear stability analysis of the flows demonstrates that viscous fingering instabilities are possible, supported essentially by the thixotropic nature of the fluid: the growth rate of unstable perturbations scales approximately with the relaxation rate λˆ−1 of the fluid, while their spatial scale is set in the radial case by (Qˆλˆ)1/2 where Qˆ is the areal injection rate, and in the fracture case by a ‘relaxation lengthscale’ which cannot be deduced by purely dimensional reasoning. It appears likely that the same instability mechanism can occur in other shear-thinning thixotropic fluids.